In this chapter, we delve into the construction of general intermediate and extrapolated spaces, essential tools in the analysis of linear operators on Banach spaces. The focus is on developing a framework that extends the classical Sobolev and extrapolation spaces for boundedly invertible operators, which will also serve as a foundation for the treatment of bi-continuous semigroups in the subsequent chapter. While many of the results presented here are well known, we aim to provide a comprehensive exposition to address cases involving non-densely defined operators, introducing new techniques and concepts where necessary. These developments are critical for the further exploration of operator theory and its applications.